This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.

Exercise 10. Show that is the th column of the matrix and that is the th entry of .

Answer: The product of the matrix and the vector is a vector, with the th entry of that vector being the inner product of the th row of with :

We then have

But by the definition of we have and for . So the above reduces to

So for all , which means that is the th column of .

We now consider the product for some . This is the inner product of the vector with the vector , which we have just shown is the th column of . We therefore have

But by the definition of we have and for . So the above reduces to

We thus have .